Breaking Instance-Independent Symmetries In Exact Graph Coloring

Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Graph coloring is also used to model more traditional CSPs relevant to AI, such as planning, time-tabling and scheduling. Provably optimal solutions may be desirable for commercial and defense applications. Additionally, for applications such as register allocation and code optimization, naturally-occurring instances of graph coloring are often small and can be solved optimally. A recent wave of improvements in algorithms for Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests generic problem-reduction methods, rather than problem-specific heuristics, because (1) heuristics may be upset by new constraints, (2) heuristics tend to ignore structure, and (3) many relevant problems are provably inapproximable. Problem reductions often lead to highly symmetric SAT instances, and symmetries are known to slow down SAT solvers. In this work, we compare several avenues for symmetry breaking, in particular when certain kinds of symmetry are present in all generated instances. Our focus on reducing CSPs to SAT allows us to leverage recent dramatic improvement in SAT solvers and automatically benefit from future progress. We can use a variety of black-box SAT solvers without modifying their source code because our symmetry-breaking techniques are static, i.e., we detect symmetries and add symmetry breaking predicates (SBPs) during pre-processing. An important result of our work is that among the types of instance-independent SBPs we studied and their combinations, the simplest and least complete constructions are the most effective. Our experiments also clearly indicate that instance-independent symmetries should mostly be processed together with instance-specific symmetries rather than at the specification level, contrary to what has been suggested in the literature

Recurrent proofs of the irrationality of certain trigonometric values

We use recurrences of integrals to give new and elementary proofs of the irrationality of pi, tan(r) for all nonzero rational r, and cos(r) for all nonzero rational r^2. Immediate consequences to other values of the elementary transcendental functions are also discussed

Effective slip over superhydrophobic surfaces in thin channels

Superhydrophobic surfaces reduce drag by combining hydrophobicity and roughness to trap gas bubbles in a micro- and nanoscopic texture. Recent work has focused on specific cases, such as striped grooves or arrays of pillars, with limited theoretical guidance. Here, we consider the experimentally relevant limit of thin channels and obtain rigorous bounds on the effective slip length for any two-component (e.g. low-slip and high-slip) texture with given area fractions. Among all anisotropic textures, parallel stripes attain the largest (or smallest) possible slip in a straight, thin channel for parallel (or perpendicular) orientation with respect to the mean flow. For isotropic (e.g. chessboard or random) textures, the Hashin-Strikman conditions further constrain the effective slip. These results provide a framework for the rational design of superhydrophobic surfaces.Comment: 4+ page

Coherent Stranski-Krastanov growth in 1+1 dimensions with anharmonic interactions: An equilibrium study

The formation of coherently strained three-dimensional islands on top of the wetting layer in Stranski-Krastanov mode of growth is considered in a model in 1+1 dimensions accounting for the anharmonicity and non-convexity of the real interatomic forces. It is shown that coherent 3D islands can be expected to form in compressed rather than in expanded overlayers beyond a critical lattice misfit. In the latter case the classical Stranski-Krastanov growth is expected to occur because the misfit dislocations can become energetically favored at smaller island sizes. The thermodynamic reason for coherent 3D islanding is the incomplete wetting owing to the weaker adhesion of the edge atoms. Monolayer height islands with a critical size appear as necessary precursors of the 3D islands. The latter explains the experimentally observed narrow size distribution of the 3D islands. The 2D-3D transformation takes place by consecutive rearrangements of mono- to bilayer, bi- to trilayer islands, etc., after exceeding the corresponding critical sizes. The rearrangements are initiated by nucleation events each next one requiring to overcome a lower energetic barrier. The model is in good qualitative agreement with available experimental observations.Comment: 12 pages text, 15 figures, Accepted in Phys.Rev.B, Vol.61, No2

Synthesis and Optimization of Reversible Circuits - A Survey

Reversible logic circuits have been historically motivated by theoretical research in low-power electronics as well as practical improvement of bit-manipulation transforms in cryptography and computer graphics. Recently, reversible circuits have attracted interest as components of quantum algorithms, as well as in photonic and nano-computing technologies where some switching devices offer no signal gain. Research in generating reversible logic distinguishes between circuit synthesis, post-synthesis optimization, and technology mapping. In this survey, we review algorithmic paradigms --- search-based, cycle-based, transformation-based, and BDD-based --- as well as specific algorithms for reversible synthesis, both exact and heuristic. We conclude the survey by outlining key open challenges in synthesis of reversible and quantum logic, as well as most common misconceptions.Comment: 34 pages, 15 figures, 2 table

Markov semigroups, monoids, and groups

A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown to be equivalent. Various interesting examples are presented, including an example of a non-Markov monoid that nevertheless admits a regular language of unique representatives over any generating set. It is shown that all finitely generated commutative semigroups are strongly Markov, but that finitely generated subsemigroups of virtually abelian or polycyclic groups need not be. Potential connections with word-hyperbolic semigroups are investigated. A study is made of the interaction of the classes of Markov and strongly Markov semigroups with direct products, free products, and finite-index subsemigroups and extensions. Several questions are posed.Comment: 40 pages; 3 figure

Parity violating cylindrical shell in the framework of QED

We present calculations of Casimir energy (CE) in a system of quantized electromagnetic (EM) field interacting with an infinite circular cylindrical shell (which we call `the defect'). Interaction is described in the only QFT-consistent way by Chern-Simon action concentrated on the defect, with a single coupling constant $a$. For regularization of UV divergencies of the theory we use % physically motivated Pauli-Villars regularization of the free EM action. The divergencies are extracted as a polynomial in regularization mass $M$, and they renormalize classical part of the surface action. We reveal the dependence of CE on the coupling constant $a$. Corresponding Casimir force is attractive for all values of $a$. For $a\to\infty$ we reproduce the known results for CE for perfectly conducting cylindrical shell first obtained by DeRaad and Milton.Comment: Typos corrected. Some references adde

Influence of Anion Nucleophilicity on Thiourea Decomposition at the Bath Chemical Deposition of PbS AND CdxPb1-xS Films

The work was financially supported by grant No 18-29-11051 and program 211 of the Government of the Russian Federation (No. 02.A03.21.0006)